Distributional, differential and integral problems: equivalence and existence results

摘要

Abstract: We are interested in studying the matter of equivalence of the following problems:Dxx(0)=f(t,x)Dg=x0(1)(1)Dx=f(t,x)Dgx(0)=x0where DxDx and DgDg stand for the distributional derivatives of xx and gg, respectively;x′g(t)x(0)=f(t,x(t)),mg-a.e.=x0(2)(2)xg′(t)=f(t,x(t)),mg-a.e.x(0)=x0where x′gxg′ denotes the gg-derivative of xx (in a sense to be specified in Section 2) and mgmg is the variational measure induced by gg; andx(t)=x0+∫t0f(s,x(s))dg(s),(3)(3)x(t)=x0+∫0tf(s,x(s))dg(s),where the integral is understood in the Kurzweil--Stieltjes sense.We prove that, for regulated functions gg, (1)(1) and (3)(3) are equivalent if ff satisfies a bounded variation assumption. The relation between problems (2)(2) and (3)(3) is described for very general ff, though, more restrictive assumptions over the function gg are required. We provide then two existence results for the integral problem (3)(3) and, using the correspondences established with the other problems, we deduce the existence of solutions for (1)(1) and (2)(2).